Morphisms of corings over different bases

A morphism (A,C)→(B,D)(A,C)\to (B,D) is a pair (α,γ)(\alpha,\gamma) where

α:A→B\alpha : A\to B is an RR-algebra morphism; by restriction this makes DD an AA-AA-bimodule by restriction. Denote also by p:D⊗AD→D⊗BDp:D\otimes_A D\to D\otimes_B D the canonical projection of bimodules induced by α\alpha.

The last two conditions can be said that the base ring extension coring B⊗AC⊗ABB\otimes_A C\otimes_A B of CC maps to DD (via map induced by γ\gamma) as a morphism of BB-corings.

Examples

Sweedler corings

The classical example of a coring is the Sweedler coring corresponding to an extension R↪SR\hookrightarrow S of unital rings. The category of descent data for this extension is equivalent to the category of comodules over the Sweedler coring.

Matrix corings

References

The notion of an AA-coring is introduced by M. Sweedler and recently lived through a renaissance in works of T. Brzeziński, R. Wisbauer, G. Böhm, L. Kaoutit, Gómez-Torrecillas, S. Caenepeel, J. Y. Abuhlail, J. Vercruysse and others, including the creation of Galois theory for corings. Some prefer to speak about AA-cocategories.